Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators.

Figure 1

Numerical study using the model function (23) for $f\left(r\right)$, with $a=0.3$. The noise intensity is $\eta =0.98696$, while the coupling strength is $K=1.5708>K_c=0.98696$. This places us well into the synchronized regime. Upper panel: Direct numerical simulation of Eq. (1), with $J=200$ and $N=5000$. The points represent the complex amplitude of each oscillator in the complex plane at the instant $t=50$, having started at time $t=0$ with all amplitudes at $A_s=0$. The red curve is the circle of unit radius. Bottom panel: Histogram of the phases of the oscillators. The thick continuous (red) curve shows a numerical computation of the steady state of the nonlinear Fokker-Planck equation (15). The thin (black) dashed curve shows the analytic estimation (24). All the distributions are normalized to the number of oscillators, that is, multiplied by the factor $2N\pi /B$, where $B=32$ is the number of bars of the histogram. The width of the histogram, also apparent in the width of the ring in the upper panel, is due to the fluctuations and to the rapid decrease of the coupling function.

Figure 2

Phase distribution for the same parameters as in Fig. 1. Histogram of the phases of the oscillators, obtained from direct numerical simulations of Eq. (1), using the same numerical data as shown in Fig. 1, but now plotting it as a histogram with five bars. The continuous curve corresponds to the steady state of the nonlinear Fokker-Planck equation (15) obtained numerically (the same as shown in Fig. 1). The squares correspond to coarse graining with $M=5$. In order to compare the results, we have used the normalization ${\sum }_{j=0}^{M-1}{P}_j\mathrm{\Delta }\varphi =2\pi N/M$, with $M=5$ (same as the number of bars) and $N=5000$, the number of oscillators considered in the simulation of Eq. (1).

Figure 3

Nullclines of the nonlinear mean field master equation (30), with $M=3$, $P_2=1-P_0-P_1$, and $\eta =1$, using the model function (23) with $a=0.3$. The solid lines (red) show $G_0(P_0,P_1)=0$ in Eq. (42) and the dashed lines (black) show $G_1(P_0,P_1)=0$. First panel: $K=1.2<{\widehat{K}}_c$, a single stable fixed point, the desynchronized state. Second panel: ${\widehat{K}}_c<K=1.56<K_c$, seven fixed points, four of them stable. Third panel: $K=1.65399=K_c$, four fixed points, one unstable and three stable. Fourth panel: $K=1.8>K_c$, three stable fixed points. The inset in the second panel clarifies the multiple crossings that contribute to the seven fixed points.

Figure 4

Equilibrium $r$ as a function of $K$, for the same parameters as in Fig. 3. Continuous branches include all the stable fixed points, while dashed branches represent the unstable fixed points.